Answer :
To solve this problem, we need to find the probability that the professor's class duration is less than 51.7 minutes, given that it follows a continuous uniform distribution between 50.0 minutes and 52.0 minutes.
Let's break it down step-by-step:
1. Understand the Uniform Distribution:
A continuous uniform distribution means every duration within the range has an equal probability of occurring. The range of our distribution is from 50.0 minutes to 52.0 minutes.
2. Identify the Components:
- The lower limit of the distribution is 50.0 minutes.
- The upper limit of the distribution is 52.0 minutes.
- We are interested in finding the probability that the class duration is less than 51.7 minutes.
3. Calculate the Probability:
In a uniform distribution, the probability of a value being less than a certain point is proportional to the length of the interval from the lower limit to that point.
The formula for the probability [tex]\( P(X < x) \)[/tex] in a continuous uniform distribution is:
[tex]\[
P(X < x) = \frac{x - \text{lower limit}}{\text{upper limit} - \text{lower limit}}
\][/tex]
Substituting the given values:
[tex]\[
P(X < 51.7) = \frac{51.7 - 50.0}{52.0 - 50.0}
\][/tex]
4. Compute the Probability:
- First, calculate the difference between 51.7 and 50.0, which is 1.7.
- Then, calculate the total range, which is 52.0 - 50.0 = 2.0.
- Finally, divide the difference by the total range:
[tex]\[
P(X < 51.7) = \frac{1.7}{2.0} = 0.85
\][/tex]
5. Conclusion:
The probability that the professor's class lasts less than 51.7 minutes is 0.85, or 85% when expressed as a percentage.
So, the answer is [tex]\( P(X < 51.7) = 0.85 \)[/tex].
Let's break it down step-by-step:
1. Understand the Uniform Distribution:
A continuous uniform distribution means every duration within the range has an equal probability of occurring. The range of our distribution is from 50.0 minutes to 52.0 minutes.
2. Identify the Components:
- The lower limit of the distribution is 50.0 minutes.
- The upper limit of the distribution is 52.0 minutes.
- We are interested in finding the probability that the class duration is less than 51.7 minutes.
3. Calculate the Probability:
In a uniform distribution, the probability of a value being less than a certain point is proportional to the length of the interval from the lower limit to that point.
The formula for the probability [tex]\( P(X < x) \)[/tex] in a continuous uniform distribution is:
[tex]\[
P(X < x) = \frac{x - \text{lower limit}}{\text{upper limit} - \text{lower limit}}
\][/tex]
Substituting the given values:
[tex]\[
P(X < 51.7) = \frac{51.7 - 50.0}{52.0 - 50.0}
\][/tex]
4. Compute the Probability:
- First, calculate the difference between 51.7 and 50.0, which is 1.7.
- Then, calculate the total range, which is 52.0 - 50.0 = 2.0.
- Finally, divide the difference by the total range:
[tex]\[
P(X < 51.7) = \frac{1.7}{2.0} = 0.85
\][/tex]
5. Conclusion:
The probability that the professor's class lasts less than 51.7 minutes is 0.85, or 85% when expressed as a percentage.
So, the answer is [tex]\( P(X < 51.7) = 0.85 \)[/tex].
